If I want 5 LED's on each side, do I have to run 5 wires plus a ground to each side or is there a way to run 2 wires to each side and then branch off from there?
You're describing
series vs. parallel circuits, if I understand you correctly. You'll probably want parallel so that you get the same voltage to all the LEDs. Basically, it will look something like:
O===O===O===O===O===[Power source]
Whereas a series would look like:
O---O---O---O---O---
|________________[Power source]
...where the top row goes from the power source to one + lead, then you wire the LEDs from + to -, then the last - back to the power source. This keeps consistent current, but distributes the voltage, so they won't be as bright. Keep in mind that this route is like Christmas tree lights (one burns out, the others lose power).
What is the best means to attach LED's to the side of a cabinet? I have a cutout in the sides with translucent plexiglass to let out the light. Are there any online resources / videos to walk a noob through this for the first time?
Drill hole, insert LED?

Think that depends on the effect you're going for. If you're going for something like a starfield or whatever, that would work; I'd hot glue them in after inserting (nothing stronger, or you'll never be able to replace them if you need/want to).
If you're trying to illuminate the plexi evenly or something, someone else might have a better idea...